scholarly journals THE GAMMA CONSTRUCTION AND ASYMPTOTIC INVARIANTS OF LINE BUNDLES OVER ARBITRARY FIELDS

2019 ◽  
pp. 1-43 ◽  
Author(s):  
TAKUMI MURAYAMA
Keyword(s):  
Line Bundles ◽  
Ground Field ◽  
Projective Varieties ◽  
Base Locus ◽  
Asymptotic Invariants ◽  
Complex Projective ◽  
Theoretic Version ◽  
Do So

We extend results on asymptotic invariants of line bundles on complex projective varieties to projective varieties over arbitrary fields. To do so over imperfect fields, we prove a scheme-theoretic version of the gamma construction of Hochster and Huneke to reduce to the setting where the ground field is $F$ -finite. Our main result uses the gamma construction to extend the ampleness criterion of de Fernex, Küronya, and Lazarsfeld using asymptotic cohomological functions to projective varieties over arbitrary fields, which was previously known only for complex projective varieties. We also extend Nakayama’s description of the restricted base locus to klt or strongly $F$ -regular varieties over arbitrary fields.

scholarly journals Bigq-ample line bundles

2012 ◽  
Vol 148 (3) ◽  
pp. 790-798 ◽  
Author(s):  
Morgan V. Brown
Keyword(s):  
Line Bundle ◽  
Coherent Sheaf ◽  
Line Bundles ◽  
Base Locus ◽  
Codimension One ◽  
Projective Scheme ◽  
Complex Projective

AbstractA recent paper of Totaro developed a theory ofq-ample bundles in characteristic 0. Specifically, a line bundleLonXisq-ample if for every coherent sheaf ℱ onX, there exists an integerm0such thatm≥m0impliesHi(X,ℱ⊗𝒪(mL))=0 fori>q. We show that a line bundleLon a complex projective schemeXisq-ample if and only if the restriction ofLto its augmented base locus isq-ample. In particular, whenXis a variety andLis big but fails to beq-ample, then there exists a codimension-one subschemeDofXsuch that the restriction ofLtoDis notq-ample.

scholarly journals Infinite-dimensional complex projective varieties

2003 ◽  
Vol 10 (2) ◽  
pp. 263-265
Author(s):  
E. Ballico
Keyword(s):  
Projective Varieties ◽  
Infinite Dimensional ◽  
Complex Projective

scholarly journals $L\sp 2\text–\overline\partial$-cohomology of complex projective varieties

1991 ◽  
Vol 4 (3) ◽  
pp. 603-603
Author(s):  
William L. Pardon ◽  
Mark A. Stern
Keyword(s):  
Projective Varieties ◽  
Complex Projective

scholarly journals On the non-divisorial base locus of big and nef line bundles on K3[2]-type varieties

2020 ◽  
pp. 1-27
Author(s):  
Ulrike Rieß
Keyword(s):  
Moduli Spaces ◽  
Hilbert Scheme ◽  
Ample Line Bundle ◽  
Base Point ◽  
K3 Surfaces ◽  
Line Bundles ◽  
Codimension Two ◽  
Base Locus ◽  
Base Loci ◽  
Symplectic Varieties

Abstract We approach non-divisorial base loci of big and nef line bundles on irreducible symplectic varieties. While for K3 surfaces, only divisorial base loci can occur, nothing was known about the behaviour of non-divisorial base loci for more general irreducible symplectic varieties. We determine the base loci of all big and nef line bundles on the Hilbert scheme of two points on very general K3 surfaces of genus two and on their birational models. Remarkably, we find an ample line bundle with a non-trivial base locus in codimension two. We deduce that, generically in the moduli spaces of polarized K3[2]-type varieties, the polarization is base point free.

scholarly journals Invariants of ample line bundles on projective varieties and their applications, II

2010 ◽  
Vol 33 (3) ◽  
pp. 416-445 ◽  
Author(s):  
Yoshiaki Fukuma
Keyword(s):  
Line Bundles ◽  
Projective Varieties

scholarly journals The standard conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of K3 surfaces

2013 ◽  
Vol 149 (3) ◽  
pp. 481-494 ◽  
Author(s):  
François Charles ◽  
Eyal Markman
Keyword(s):  
Hilbert Scheme ◽  
K3 Surfaces ◽  
Cohomology Algebra ◽  
Hilbert Schemes ◽  
Projective Varieties ◽  
Hilbert Scheme Of Points ◽  
Complex Projective ◽  
Symplectic Varieties

AbstractWe prove the standard conjectures for complex projective varieties that are deformations of the Hilbert scheme of points on a K3 surface. The proof involves Verbitsky’s theory of hyperholomorphic sheaves and a study of the cohomology algebra of Hilbert schemes of K3 surfaces.

scholarly journals On Analytic Todd Classes of Singular Varieties

2019 ◽  
Author(s):  
Francesco Bei ◽  
Paolo Piazza
Keyword(s):  
Fundamental Group ◽  
Complex Space ◽  
Homology Class ◽  
Classifying Space ◽  
Projective Varieties ◽  
Singular Varieties ◽  
Push Forward ◽  
Birational Invariant ◽  
Complex Projective

Abstract Let $(X,h)$ be a compact and irreducible Hermitian complex space. This paper is devoted to various questions concerning the analytic K-homology of $(X,h)$. In the 1st part, assuming either $\dim (\operatorname{sing}(X))=0$ or $\dim (X)=2$, we show that the rolled-up operator of the minimal $L^2$-$\overline{\partial }$ complex, denoted here $\overline{\eth }_{\textrm{rel}}$, induces a class in $K_0 (X)\equiv KK_0(C(X),\mathbb{C})$. A similar result, assuming $\dim (\operatorname{sing}(X))=0$, is proved also for $\overline{\eth }_{\textrm{abs}}$, the rolled-up operator of the maximal $L^2$-$\overline{\partial }$ complex. We then show that when $\dim (\operatorname{sing}(X))=0$ we have $[\overline{\eth }_{\textrm{rel}}]=\pi _*[\overline{\eth }_M]$ with $\pi :M\rightarrow X$ an arbitrary resolution and with $[\overline{\eth }_M]\in K_0 (M)$ the analytic K-homology class induced by $\overline{\partial }+\overline{\partial }^t$ on $M$. In the 2nd part of the paper we focus on complex projective varieties $(V,h)$ endowed with the Fubini–Study metric. First, assuming $\dim (V)\leq 2$, we compare the Baum–Fulton–MacPherson K-homology class of $V$ with the class defined analytically through the rolled-up operator of any $L^2$-$\overline{\partial }$ complex. We show that there is no $L^2$-$\overline{\partial }$ complex on $(\operatorname{reg}(V),h)$ whose rolled-up operator induces a K-homology class that equals the Baum–Fulton–MacPherson class. Finally in the last part of the paper we prove that under suitable assumptions on $V$ the push-forward of $[\overline{\eth }_{\textrm{rel}}]$ in the K-homology of the classifying space of the fundamental group of $V$ is a birational invariant.

scholarly journals Seshadri Constants for Curve Classes

2019 ◽  
Author(s):  
Mihai Fulger
Keyword(s):  
Normal Bundle ◽  
Projective Variety ◽  
Smooth Curve ◽  
Smooth Projective Variety ◽  
Projective Varieties ◽  
Seshadri Constants ◽  
Base Locus ◽  
Nef Divisors

Abstract We develop a local positivity theory for movable curves on projective varieties similar to the classical Seshadri constants of nef divisors. We give analogues of the Seshadri ampleness criterion, of a characterization of the augmented base locus of a big and nef divisor, and of the interpretation of Seshadri constants as an asymptotic measure of jet separation. As application, we show in any characteristic that if $C$ is a smooth curve with ample normal bundle in a smooth projective variety then the class of $C$ is in the strict interior of the Mori cone. This was conjectured by Peternell and proved by Ottem and Lau in Characteristic 0.

Real algebraic cycles on complex projective varieties

1997 ◽  
Vol 225 (2) ◽  
pp. 177-198 ◽  
Author(s):  
Joost van Hamel
Keyword(s):  
Algebraic Cycles ◽  
Projective Varieties ◽  
Complex Projective
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